Sensitive dependence on initial conditions (S.I.C.)
This is a more powerful definition than you find in most books. It
is published in R.M. Corless,
``Continued Fractions and Chaos'', Proceedings of the Organic
Mathematics Workshop.
Suppose
our problem is

ie. , subject to

We ignore the boundary conditions on . If there exist
constants M > 0 and (M not too large, not too small) such that for every
there exists a
with such that the
solution of

has for some then we say that the
problem

is S.I.CRemark: This definition says that some perturbations of
the initial condition lead
to finite separation
after an exponentially short length of time -- that is, initial
errors may grow exponentially,
for a short while.

The weasel words ``M not too large, not too small'' are
there for practicality. Notice
the limit is not involved. This makes it a
different definition than the
others in the literature.